This paper reviews some earlier results of the authors given in the papers by the second and third author [J. Difference Equ. Appl. 7, No. 1, 21–50 (2001; Zbl 0972.39004)], and the first and the third author [Dyn. Syst. Appl. 12, No. 1–2, 23–43 (2003; Zbl 1053.39029)]. They address linear dynamic equations on time scales \({\mathbb T}\), which include ordinary differential equations (\({\mathbb T}={\mathbb R}\)), ordinary difference equations (\({\mathbb T}={\mathbb Z}\)) or \(q\)-difference equations (\({\mathbb T}=\{q^n\}_{n\geq 0}\), \(q>1\)) as special cases. More detailed, the following results are discussed:
(1) If the linear dynamic equation \[ y^\Delta=A(t)y \tag{*} \] in \({\mathbb R}^n\) possesses an ordinary dichotomy, then there exists a homeomorphism between the bounded solutions of \((\ast)\) and of the perturbed equation \[ x^\Delta=[A(t)+R(t)]x, \] provided the perturbation matrix \(R(t)\) is of class \(L^1\) w.r.t. the \({\mathbb T}\)-integral.
(2) In order to apply this result to the special case of a diagonal matrix \(A(t)= \text{diag}(\lambda_1(t),\ldots,\) \(\lambda_n(t))\), sufficient criteria for the existence of an ordinary dichotomy are given in terms of the functions \(\lambda_i(t)\). They are illustrated for the classical examples \({\mathbb T}={\mathbb R}\) and \({\mathbb T}={\mathbb Z}\).
(3) The authors discuss possible strategies to transform equation \((\ast)\) into \[ x^\Delta=[\Lambda(t)+R(t)]x \] with a diagonal matrix \(\Lambda(t)\) using an invertible linear transformation \(P(t)\). This yields a representation \(Y(t)=P(t)[I+o(1)]D(t)\) for \(t\to\infty\) of a fundamental matrix for \((\ast)\), with the Hardy-Littlewood symbol \(o(1)\) and a diagonal matrix \(D(t)\). Such an approach is illustrated for the special case \(A(t)=C+R(t)\) with a diagonalizable constant matrix \(C\). Then it is possible to establish a representation of \(Y(t)\) as above.
Finally, results on the asymptotic behavior of the scalar exponential function on time scales from Bodine and Lutz [loc cit] conclude the paper.